7 SEMICONTINUOUS FUNCTIONS 31 (b) Deduce from (a) that there are the following three alternatives : either the series (ane~^ns) does not converge for any value of s, or it converges for all values of s, or else there exists a real number cr0 such that the series converges for &s > a0 and does not converge when &s < cr0 - In the first (resp. second) case we put cr0 = +00 (resp. — oo). In all cases, the (extended real) number cr0 is called the abscissa of convergence of the series. The sum of the series is an analytic function of s in the region %s > a0 . (c) Show that if o-0 ^0, then <TO = lim sup (log |S0>n|)/An. (Show first that if the JI-K» series with general term ane~*nS — bn is convergent (with s real and >0), then we have |S0,ii| ^ K>A"S where K is a constant, by writing an = bne*"s. Then show that if y = lim sup (log |S0, n|)/An, the series (ane~*nS) converges when s = y + S, where 8 is n~>oo real and >0, by arguing as in (a).) (d) Let CTI be the abscissa of convergence of the Dirichlet series with general term \an\e~*nS. Then a± ^ o-0 . If a0 < + oo, show that ^ ,. 1°SW (TI — o"0 ^ lim sup — r — . n-*oo An (Remark that, given e > 0, we have \an\ <; e(<T°+e)An for all sufficiently large n.) Consider the case where An = n (cf. Section 9.1 , Problem 1). Show that, for the series with general term (—IV : _ IL. g-siogiogn Vn we have a0 — — oo and crt = +00. 10. Let/be a real-valued function defined on the interval [0, + oo [, satisfying the following conditions: (D /(J+0£/(*)+/(0; (2) there exists a number M > 0 such that |/(r)| ^ M/ for all /. Show that the limits r R r /(O lim — , p = lim -1- t-*0 t t-t' + oo t exist and are finite, and that a/ <i/(0 < fit for all t ;> 0. (To establish the existence of a it is enough to show that, if we take a — lim sup /(/)// (Problem 8), then I-+0 a ^f(t)/t for all t > 0. Take a decreasing sequence (tn) tending to 0, such that \imf(tn)/tn = a, and let kn be the integral part of tjtn . Show that The proof of the existence of lim f(t)/t is similar; put ft = lim ii t-* + °o r-* + oo 11. Let/be a continuous mapping of an open set U in a Banach space E into a Banach space F. For each x e U, let \\y~X\\ y-*,y** y~ We have 0 g D~/(x)For each